The goal of this talk, based on a joint work with Lucas Rey, is to prove a full classification of 2D-Ising models defined on isoradial graphs, frustrated or not, whose underlying spectral curve has genus 1. As a specific case, we recover Baxter's Z-invariant Ising model, thus extending his class of models to real coupling constants. In doing so, we identify the different behaviors of phase transition observed in the physics literature, using the underlying spectral curve. While in the case of positive coupling constants the curve is known to be Harnack, we here identify two kinds of non-Harnack curves that occur in frustrated Ising models. We prove that in all cases the curve is maximal, and undergoes an algebraic phase transition, in the sense that the genus goes from 1 to 0, thus shedding light on the physics phase transition. Our results also provide a natural framework for a further systematic study of the frustrated Ising model, amenable to proving local formulas. Note that in the course of the talk, we will define isoradial graphs and spectral curves.